University of Virginia
Physics Department

## Energy of a Pendulum

A Physical Science Activity

2003 Virginia SOLs

• PS.1
• PS.6

Objectives

Students will

• construct a pendulum;
• explain how a pendulum behaves;
• describe the potential and kinetic energy of a pendulum.

Motivation for Learning

Demonstration of a Pendulum Swing

Materials

• Heavy duty string
• 1-kg mass

Procedure

1. Tie a piece of heavy-duty string to a support on the ceiling (or someplace high with clearance below). The string should be long enough to reach from the ceiling to your waist after it is tied.
2. Tie a 1-kg mass to the end of the string, raise the mass and let it swing back and forth. Explain to the students that this is an example of a pendulum.
3. Demonstrate that the mass will not move unless you lift it . When you lift the mass (or any object) you are giving it energy. This type of energy is called potential energy. The higher you lift it the more potential energy you give it. When you release the mass it moves down and the potential energy you gave it starts changing into kinetic energy . There is always the same amount of energy (neglecting friction), it just changes from potential to kinetic. There is kinetic energy as long as there is movement of the mass.
4. Pull the mass back with the string taut, hold it to your nose and release it. It will swing away from you but never quite return to the original position. This takes some courage, because as you look at the mass swinging back towards you, it appears that the mass will hit you in the nose. Be careful not to give the mass a push when you initially release it, or it may hit you in the nose! Air friction robs some of the energy from the pendulum and it can never swing back to its original height. Invite the students to try holding the mass to their nose, releasing it and standing still as the bob swings back toward their nose. Caution them not to give the mass a push!

Background Information

The gravitational potential energy of an object is related to its mass and the vertical height to which the object is lifted. It can be expressed by the following formula:

 G.P.E.= Weight X Height = mgh

You can see from the formula that the greater the weight and the higher the position of the ball, the greater the potential energy. The kinetic energy of an object in motion is related to the mass of object (m) and its velocity (v). This mathematical relationship is expressed as:

 K.E. = 1/2 mv2

A simple pendulum is an example of simple harmonic motion. It contiunes swinging back and forth. During this swinging, there is constant exchange between potential and kinetic energy. When a pendulum is the farthest up in its swing, it is at its maximum height which gives it maximum potential energy. At this same point in the swing, the pendulum is motionless for a fraction of a second because it is changing direction. Since the speed is zero, the kinetic energy is also zero. After the pendulum changes direction and begins swinging again, it is losing height, but gaining speed. This also means that the pendulum is losing potential energy, but gaining kinetic energy. When the pendulum reaches the very bottom of the swing, it is at its lowest point and greatest speed. This means that the pendulum has zero potential energy (with respect to its rest position) and maximum kinetic energy. This pattern continues for the rest of the swings. Ignoring air resistance, the total energy of the system (G.P.E. + K.E) should be the same at every position in the swing.

This student activity is designed to have the students determine that the greatest kinetic energy occurs at the very bottom of the swing. They do this by having the pedulum hit a block and determine which part of the swing causes the block to move the most. Students should understand that this is a qualitative demonstration of energy transfer. They should not expect to get precise data, but should be able to identify the points of high potential and kinetic energies, and how the total energy is conserved.

### Student Activity

Where is the mass at the end of a pendulum moving the fastest?

Materials

• String about 1 meter long
• 1-kg mass with hook at the top
• 2-3 books
• Small wooden block approximately (5cm X 5cm X 5cm)
• Metric ruler

Procedure

1. Make a pendulum by tying one end of a piece of string to a hook on a 1 kg mass. This mass is called the pendulum bob. Hang the pendulum over the end of the edge of a desk or table so that it just clears the floor.
2. Tape the upper end of the string to the side of the desk so that the pendulum can swing freely. Swing the pendulum to be sure that it does not touch the floor.
3. Place a block of wood under the pendulum. The bob of the pendulum should hit the block at the bottom of its first swing. Mark the spot where the block will be with a piece of tape.
4. Keeping the string taught, raise the pendulum to a point 30 cm above the floor. You will need a ruler to measure this distance from the floor to the bob. Keeping the bob still, find the spot on the floor directly underneath the bob, mark it, and measure the distance from the spot under the bob to the wooden block. Enter the distance in the 1st data table below.
5. Release the bob and measure the distance that the block moved from the tape. Enter the distance in the table and repeat the experiment three times. You will need to find the average distance and enter it in the 1st data table.
6. Again, hold the bob 30 cm above the floor. Move the wooden block horizontally towards the bob, so that the bob will hit the block at about 1/2 the angle that the bob hit the block previously (see the diagram). Place the wooden block on top of 1 or 2 books so that the bob will still hit the block. Measure the new distance from the wooden block to the bob and enter it in table #2.
7. Release the bob from the 30-cm height and let it hit the block. Repeat the experiment three times and find the average as before. Enter all measurements in table #2.

Data Sheet

Table 1
 Trial Distance From Block to Bob Distance Block Moves 1 2 3
Average Distance Block Moved ____________________________

Table 2
 Trial Distance From Block to Bob Distance Block Moves 1 2 3
Average Distance Block Moved ____________________________

1. What happens to the block of wood when it was hit by the pendulum bob?

2. Kinetic energy is energy of motion and potential energy is stored energy. What kind of energy did the bob give to the block?

3. Where would you place the block for the pendulum bob to push the block the farthest?

4. If the energy of the pendulum is either stored (potential) or moving (kinetic), where in the swing does the pendulum have the most potential energy? Explain how you know.

5. Where in the swing does the pendulum have the most kinetic energy (speed)? Explain how you know.

Students with special needs

Assessment

A roller coaster is very similar to a pendulum in terms of kinetic and potential energy. Use the illustration below to answer the questions.

1. __________ Which letter in the diagram represents the place where the cars on the roller

coaster would have the most potential energy?

2. __________ Which letter represents the place where the cars would have the most kinetic

energy?

3. __________ Which letter represents the place where the cars would have the least kinetic

energy?

4. __________ Which letter represents the place where the energy would be changing from

kinetic to potential energy?

5. __________ Which letter represents the place where the energy would be changing from potential to kinetic energy?

6. __________ Where in the diagram would the cars be moving with the greatest speed?

7. Why do you think that the first hill on the roller coaster must be the tallest? (explain your answer in terms of kinetic and potential energy.